Let X be a real strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm and C be a nonempty closed convex subset of X . Let { T n } n = 1 ∞ be a sequence of nonexpansive self-mappings on C such that the common fixed point set F ≔ ⋂ n = 1 ∞ F ( T n ) ≠ 0̸ and f : C → C be a given contractive mapping, and { λ n } be a sequence of nonnegative numbers in [ 0 , 1 ] . Consider the following relaxed viscosity approximation method { x n + 1 = ( 1 − α n − β n ) x n + α n f ( y n ) + β n W n y n , y n = ( 1 − γ n ) x n + γ n W n x n , n ≥ 1 where W n is the W -mapping generated by T n , T n − 1 , … , T 1 and λ n , λ n − 1 , … , λ 1 for each n ≥ 1 . It is proven that under very mild conditions on the parameters, the sequence { x n } of approximate solutions generated by the proposed method converges strongly to some p ∈ F where p is the unique solution in F to the following variational inequality: 〈 ( I − f ) p , J ( p − x ∗ ) 〉 ≤ 0 , ∀ x ∗ ∈ F .